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Homotopy classification and the generalized Swan homomorphism
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- Journal:
- Journal of K-Theory / Volume 4 / Issue 3 / December 2009
- Published online by Cambridge University Press:
- 07 January 2009, pp. 491-536
- Print publication:
- December 2009
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- Article
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In his fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism
for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [18], [21]). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form 
here 
is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → J → Fn → … → F0 → F0 → M → 0 where each Fr is finitely generated free.
for any finite group 
here 
is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → 